Analog Intuition

Part 1 · Impedance Matching

Need for Matching

One idea: for a resistive source, the load gets the most power when RL = RS.

Power into RL

Fixed source RS and open-circuit voltage VS · vary the load

V_load —
P_load —
P_available —
Fraction delivered —
\[ V_L = V_S\,\dfrac{R_L}{R_S + R_L} \]
= —
\[ P_L = \dfrac{1}{2}\,\dfrac{|V_L|^2}{R_L} \]
= —
\[ P_{\mathrm{avail}} = \dfrac{V_S^2}{8\,R_S} \]
= —

Peak source \(V_S\); average power into \(R_L\). Maximum when \(R_L = R_S\).

Try it: drag RL across the peak. Half the available power is lost in RS even at the match — but less reaches the load if RL is far from RS.

The one idea

A source with internal resistance \(R_S\) can deliver at most a certain power to a load. For purely resistive ports, that maximum is at \(R_L = R_S\). Move away from that point and the load power falls — even if the source voltage is huge.

\[P_L = \frac{1}{2}\frac{|V_L|^2}{R_L},\qquad V_L = V_S\frac{R_L}{R_S+R_L},\qquad P_{\mathrm{avail}}=\frac{V_S^2}{8R_S}\]

Why RF people obsess about “50 Ω”

Cables, instruments, and many ICs are standardized to a real impedance (often 50 Ω). If your circuit presents something very different, you bounce energy (Part 3) and lose power (this plot). Matching networks exist to present the right resistance (and later, the right reactance) to the source.

What we ignored (on purpose)

Real loads are complex: ZL = R + jX. The full rule is conjugate match: ZL = ZS*. Parts 2–5 build the language for that. Part 6 puts it on the Smith chart tool.